Origin of the equation and its name

There are two equivalent forms of the sine-Gordon equation. In the (real) space-time coordinates, denoted (x,t), the equation reads

, phi_{tt}- phi_{xx} + sinphi = 0.

Passing to the light cone coordinates (u, v), akin to asymptotic coordinates where

u=frac{x+t}2, quad v=frac{x-t}2,

the equation takes the form:

varphi_{uv} = sinvarphi.

This is the original form of the sine-Gordon equation, as it was considered in the nineteenth century in the course of investigation of surfaces of constant Gaussian curvatureK = −1, also called pseudospherical surfaces. Choose a coordinate system for such a surface in which the coordinate mesh u = const, v = const is given by the asymptotic lines parameterized with respect to the arc length. The first fundamental form of the surface in these coordinates has a special form

ds^2 = du^2 + 2cosvarphi du dv + dv^2,

where φ expresses the angle between the asymptotic lines, and for the second fundamental form, L = N = 0. Then the Codazzi-Mainardi equation expressing a compatibility condition between the first and second fundamental forms results in the sine-Gordon equation. The study of this equation and of the associated transformations of pseudospherical surfaces in the 19th century by Bianchi and Bäcklund led to the discovery of Bäcklund transformations.

1-soliton solutions

The 1-soliton solution for which we have chosen the positive root for gamma is called a kink, and represents a twist in the variable phi which takes the system from one solution phi=0 to an adjacent with phi=2pi. The states phi=0(textrm{mod}2pi) are known as vacuum states as they are constant solutions of zero energy. The 1-soliton solution in which we take the negative root for gamma is called an antikink. The form of the 1-soliton solutions can be obtained through application of a Bäcklund transform to the trivial (constant vacuum) solution and the integration of the resulting first-order differentials:

The 1-soliton solutions can be visualized with the use of the elastic ribbon sine-Gordon model as discussed by Dodd and co-workers. Here we take a clockwise (left-handed) twist of the elastic ribbon to be a kink with topological charge vartheta_{textrm{K}}=-1. The alternative counterclockwise (right-handed) twist with topological charge vartheta_{textrm{AK}}=+1will be an antikink.

2-soliton solutions

Multi-soliton solutions can be obtained through continued application of the Bäcklund transform to the 1-soliton solution, as prescribed by a Bianchi lattice relating the transformed results. The 2-soliton solutions of the sine-Gordon equation show some of the characteristic features of the solitons. The traveling sine-Gordon kinks and/or antikinks pass through each other as if perfectly permeable, and the only observed effect is a phase shift. Since the colliding solitons recover their velocity and shape such kind of interaction is called an elastic collision.

Another interesting 2-soliton solutions arise from the possibility of coupled kink-antikink behaviour known as a breather. There are known three types of breathers: standing breather, traveling large amplitude breather, and traveling small amplitude breather.

3-soliton solutions

3-soliton collisions between a traveling kink and a standing breather or a traveling antikink and a standing breather results in a phase shift of the standing breather. In the process of collision between a moving kink and a standing breather,
the shift of the breather Delta_{textrm{B}} is given by:

where v_{textrm{K}} is the velocity of the kink, and omega is the breather's frequency. If the old position of the standing breather is x_{0}, after the collision the new position will be x_{0}+Delta_{textrm{B}}.

Related equations

Another closely related equation is the elliptic sine-Gordon equation, given by

varphi_{xx} + varphi_{yy} = sinvarphi,

where φ is now a function of the variables x and y. This is no longer a soliton equation, but it has many similar properties, as it is related to the sine-Gordon equation by the analytic continuation (or Wick rotation) y = it.

Quantum version

In quantum field theory the sine-Gordon model contains a parameter. The particle spectrum consists of a soliton, an anti-soliton and a finite (possibly zero) number of breathers. The number of the breathers depends on the value of the parameter.

In finite volume and on a half line

On can also consider the sine-Gordon model on a circle, on a line segment, or on a half line. It is possible to find boundary conditions which preserve the integrability of the model. On a half line the spectrum contains boundary bound states in addition to the solitons and breathers.

Supersymmetric sine-Gordon model

A supersymmetric extension of the sine-Gordon model also exists. Integrability preserving boundary conditions for this extension can be found as well.